A168546 -id:A168546 - cp's OEIS frontend

A215668 Decimal expansion of the zero z in (0,Pi/2) of the function sin(sin(x))/x - cos(cos(x))/x.

6, 9, 2, 7, 2, 8, 5, 7, 0, 1, 8, 6, 8, 3, 3, 8, 8, 8, 3, 4, 7, 3, 6, 5, 2, 2, 0, 5, 8, 0, 2, 9, 4, 0, 3, 0, 2, 3, 6, 7, 7, 4, 5, 4, 4, 8, 3, 7, 8, 5, 5, 4, 6, 6, 2, 5, 0, 4, 6, 4, 2, 6, 7, 6, 4, 5, 3, 9, 7, 4, 2, 4, 9, 5, 6, 6, 1, 6, 4, 1, 6, 7, 4, 4, 3, 9, 5, 2, 8, 3, 9, 5, 2, 2, 1, 4, 7, 2, 8, 7, 7, 8, 5, 9, 8, 6, 5, 5, 8

Offset: 0

Roman Witula, Aug 20 2012

It is proved (see Witula et al.'s reference) that the function h(x) := sin(sin(x))/x - cos(cos(x))/x is decreasing in the interval (0,Pi/2) and has zero z in (0,Pi/4). We have sin(sin(z))/z = cos(cos(z))/z = 0.933396189408898411846964. Moreover the following relation hold: F(z) = min{F(x): x \in R} = 0.10712694487, where F(x) := cos(sin(x)) - sin(cos(x)) - see also A215670 and the Witula et al.'s reference for more information.

			z = 0.692728570186833888... rad  (= 39.6904234198376057055880... deg).

R. Witula, D. Jama, E. Hetmaniok, D. Slota, On some inequality of the trigonometric type, Zeszyty Naukowe Politechniki Slaskiej - Matematyka-Fizyka (Science Fascicle of Silesian Technical University - Math.-Phys.), 92 (2010), 83-92.

Cf. A215670, A215832, A215833, A168546, A216891.

A215670 Decimal expansion of the min value of F(x) := cos(sin(x)) - sin(cos(x)), x in R.

Original entry on oeis.org

1, 0, 7, 1, 2, 6, 9, 4, 4, 8, 7, 2, 9, 5, 2, 9, 9, 6, 1, 1, 2, 0, 2, 9, 4, 8, 1, 3, 4, 7, 4, 1, 9, 1, 7, 4, 8, 4, 3, 3, 2, 1, 3, 9, 8, 2, 6, 3, 3, 6, 6, 1, 2, 8, 9, 0, 4, 4, 7, 3, 5, 5, 8, 4, 2, 6, 4, 7, 9, 8, 6, 2, 7, 2, 1, 1, 3, 1, 1, 6, 9, 6, 6, 8, 5, 8, 5, 1, 8, 7, 7, 9, 6, 2, 3, 5, 4, 7, 3, 7, 5, 9, 2, 3

Offset: 0

Roman Witula, Aug 20 2012

We note that dF(x)/dx = (-1/2)*h(x)*sin(2*x), x in (0,Pi/2), where h(x) is the function discussed in comments to A215668 (see also Witula et al.'s reference for more informations).

			min{F(x): x in R} = F(z) = 0.1071269448729529961...

R. Witula, D. Jama, E. Hetmaniok, D. Slota, On some inequality of the trigonometric type, Zeszyty Naukowe Politechniki Slaskiej - Matematyka-Fizyka (Science Fascicle of Silesian Technical University - Math.-Phys.), 92 (2010), 83-92.

Cf. A215668, A215832, A215833, A168546, A216891.

F(z) = cos(sin(z)) - sin(cos(z)) = (cos(z) - sin(z))*(cos(cos(z)) + sin(sin(z)))*cos(cos(z))/(cos(sin(z)) + sin(cos(z)))*cos(z) = cos(2*z)*cos(cos(z))^2/(cos(sin(z)) + sin(cos(z)))*cos(z)^2 = (1 - tan(z)^2)*cos(cos(z))^2/(cos(sin(z)) + sin(cos(z))), where z := A215668.

A215832 Decimal expansion of the maximum of the function f(x) = log(cos(sin(x)))/log(sin(cos(x))), x in (0,Pi/2).

Original entry on oeis.org

6, 4, 1, 0, 1, 9, 2, 3, 7, 6, 3, 2, 7, 9, 0, 3, 5, 5, 6, 8, 8, 8, 4, 6, 8, 6, 8, 8, 1, 6, 6, 2, 4, 2, 4, 1, 9, 6, 8, 9, 4, 4, 5, 6, 3, 2, 5, 5, 8, 1, 4, 2, 0, 6, 7, 6, 6, 3, 0, 5, 2, 8, 9, 8, 7, 2, 2, 4, 1, 1, 1, 9, 7, 6, 8, 8, 3, 9, 5, 6, 4, 2, 0, 0, 9, 2, 0, 9, 7, 6, 8, 4, 8, 0, 2, 8, 4, 3, 4, 6, 9, 4, 0, 7, 4, 3, 8, 6, 5, 1, 1, 7, 8, 2, 4, 7, 1, 0, 0, 5, 0, 4, 1, 3, 4

Offset: 0

Roman Witula, Aug 24 2012

The inverse of this maximum is equal to A215833. The argument z in (0,Pi/2) for which f(z) = max{f(x): x in (0,Pi/2)} is given in A168546. We note that f is increasing in the interval (0,z) and decreasing in the interval (z,Pi/2).

			We have M := max{f(x): x in (0,Pi/2)} = 0.6410192376327.

R. Witula, D. Jama, E. Hetmaniok, D. Slota, On some inequality of the trigonometric type, Zeszyty Naukowe Politechniki Slaskiej - Matematyka-Fizyka (Science Fascicle of Silesian Technical University - Math.-Phys.), 92 (2010), 83-92.

Cf. A215833, A168546, A215670, A215668, A216891.

A215833 Decimal expansion of the maximum value p>0, such that (cos(sin(x)))^p >= sin(cos(x)), x in (0,Pi/2).

Original entry on oeis.org

1, 5, 6, 0, 0, 1, 5, 5, 8, 3, 4, 5, 2, 5, 0, 9, 0, 6, 1, 0, 7, 6, 6, 5, 3, 4, 8, 5, 2, 5, 9, 1, 1, 3, 5, 6, 0, 8, 7, 7, 3, 9, 1, 9, 0, 8, 6, 8, 7, 8, 4, 5, 1, 2, 0, 1, 2, 7, 6, 6, 1, 6, 2, 1, 3, 7, 8, 4, 4, 0, 9, 9, 4, 3, 6, 2, 7, 8, 4, 6, 3, 9, 7, 2, 7, 9, 9, 1, 5, 8, 1, 0, 3, 9, 7, 2, 2, 2, 2, 2, 9, 0, 9, 4, 6, 2, 6, 0, 8, 1, 2, 6, 3, 7, 7, 3, 1, 7, 3, 9, 0, 7, 4, 9, 8

Offset: 1

Roman Witula, Aug 24 2012

Equal to the inverse of the maximum of the function f(x) from A215832.

R. Witula, D. Jama, E. Hetmaniok, D. Slota, On some inequality of the trigonometric type, Zeszyty Naukowe Politechniki Slaskiej - Matematyka-Fizyka (Science Fascicle of Silesian Technical University - Math.-Phys.), 92 (2010), 83-92.

Cf. A215832, A168546, A215670, A215668, A216891.

max p = 1/A215832 = 1.5600155834525...

A216891 Decimal expansion of the minimal zero x(1) of the function F(x) = f(f(x)) - x, where f(x) = cos(x) - sin(x).

Original entry on oeis.org

8, 3, 0, 1, 9, 8, 5, 1, 7, 0, 6, 7, 8, 2, 3, 9, 3, 4, 5, 5, 2, 2, 5, 6, 2, 7, 1, 9, 5, 5, 2, 7, 1, 0, 6, 5, 7, 7, 8, 2, 0, 6, 3, 0, 8, 4, 3, 9, 4, 5, 4, 3, 7, 3, 1, 9, 3, 9, 5, 5, 2, 4, 1, 2, 2, 1, 6, 0, 8, 4, 8, 3, 2, 0, 4, 5, 6, 1, 8, 8, 9, 6, 2, 2, 6, 4, 1, 6, 3, 8, 6, 9, 7, 2, 6, 2, 9, 1, 2, 1, 5, 9, 1, 2, 3

Offset: 0

Roman Witula, Sep 19 2012

Let x(2) and x(3) denote the remaining zeros of F(x), x(2) < x(3). Then it could be proved that f(x(1)) = x(3), f(x(3)) = x(1), and f(x(2)) = x(2).
The decimal expansions of x(2) and x(3) in A206291 and A216863 respectively are presented.
We note that the plot of the restriction of F(x) to the interval [-2,2] "is very similar" to the plot of the polynomial (x-x(1))*(x-x(2))*(x-x(3)) for x in [-2,2].
Let A = {x in R: f^n(x) = x(2) for some nonnegative integer n, where f^n denotes the n-th iteration of f}. Then if z is a real number, which does not belong to A, and z(0):= z, z(n+1) = f(z(n)) = sqrt(2)*sin(Pi/4 - z(n)), n in N, then one of the subsequences either {z(2*n-1)} or {z(2*n)} is convergent to x(1) and the second one is convergent to x(3).

			We have x(1) = -0.830198517...

R. Witula, D. Slota and Szeged Problem Group "Fejentalaltuka", An Iteration Convergence: 11318[2007,745], Amer. Math. Monthly, 116 No 7 (2009), 648-649.

Cf. A216863, A206291, A215668, A215670, A215832, A215833, A168546.

A206291 Decimal expansion of the middle zero x(2) of the function F(x) = f(f(x)) - x, where f(x) = cos(x) - sin(x).

Original entry on oeis.org

4, 5, 6, 6, 2, 4, 7, 0, 4, 5, 6, 7, 6, 3, 0, 8, 2, 4, 4, 3, 7, 6, 9, 7, 4, 5, 7, 1, 2, 8, 4, 5, 7, 3, 7, 5, 8, 9, 8, 2, 3, 1, 6, 1, 3, 8, 9, 2, 2, 5, 6, 3, 2, 5, 2, 4, 2, 2, 7, 8, 2, 3, 0, 7, 7, 3, 4, 5, 3, 8, 6, 2, 9, 0, 0, 6, 4, 2, 2, 9, 7, 0, 2, 9, 1, 4, 3, 1, 8, 9, 9, 3, 0, 1, 7, 7, 1, 5, 8, 8, 6, 8, 4, 6, 1

Offset: 0

Roman Witula, Sep 22 2012

The decimal expansions of the other two zeros (x(1) and x(3)) of F(x) are A216891 and A216863, respectively. See the Comments at A216891 for more information on the properties of these zeros.

			We have f(x(2)) = x(2) = 0.45662470456763..., sin(x(2)) = 0.4409211... and x(2) - sin(x(2)) > 0.015703...

R. Witula, D. Slota and Szeged Problem Group "Fejentalaltuka", An Iteration Convergence: 11318[2007,745], Amer. Math. Monthly, 116, No. 7 (2009), 648-649.

Cf. A216891, A216863, A215668, A215670, A215832, A215833, A168546.

A216863 The decimal expansion of the maximal zero x(3) of the function F(x) = f(f(x)) - x, where f(x) = cos(x) - sin(x).

Original entry on oeis.org

1, 4, 1, 2, 7, 9, 4, 5, 8, 5, 7, 2, 7, 6, 2, 2, 2, 4, 5, 2, 9, 3, 5, 7, 7, 5, 8, 6, 5, 0, 6, 6, 3, 7, 6, 3, 2, 1, 2, 4, 5, 5, 8, 8, 9, 1, 2, 7, 3, 8, 1, 5, 1, 6, 5, 8, 6, 4, 1, 7, 7, 5, 2, 5, 5, 3, 0, 1, 1, 2, 5, 2, 4, 7, 7, 1, 2, 2, 0, 5, 5, 1, 8, 2, 4, 3, 3, 4, 0, 8, 7, 8, 5, 0, 0, 6, 0, 8, 9, 2, 5, 6, 1, 5, 0

Offset: 1

Roman Witula, Sep 18 2012

The decimal expansions of the only other zeros x(1) and x(2) of F(x) are given in A216891 and A206291. See the comments in A216891 for more information about intriguing properties of these zeros.

			We have x(3) = 1.4127945857276222... < sqrt(2).

R. Witula, D. Slota and Szeged Problem Group "Fejentalaltuka", An Iteration Convergence: 11318[2007, 745], Amer. Math. Monthly, 116 No 7 (2009), 648-649.

Cf. A216891, A206291, A215668, A215670, A215832, A215833, A168546.

f[x_] := Cos[x] - Sin[x];
FindRoot[f[f[x]] - x, {x, 1.4}, WorkingPrecision -> 110] (* T. D. Noe, Sep 24 2012; first line by Rick L. Shepherd, Jan 04 2014 *)

default(realprecision,110);
f(x) = cos(x) - sin(x);
solve(x = 1.4, 1.5, f(f(x)) - x) \\ Rick L. Shepherd, Jan 03 2014

Offset corrected by Rick L. Shepherd, Jan 03 2014

cp's OEIS Frontend

A215668 Decimal expansion of the zero z in (0,Pi/2) of the function sin(sin(x))/x - cos(cos(x))/x.

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A215670 Decimal expansion of the min value of F(x) := cos(sin(x)) - sin(cos(x)), x in R.

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A215832 Decimal expansion of the maximum of the function f(x) = log(cos(sin(x)))/log(sin(cos(x))), x in (0,Pi/2).

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A215833 Decimal expansion of the maximum value p>0, such that (cos(sin(x)))^p >= sin(cos(x)), x in (0,Pi/2).

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A216891 Decimal expansion of the minimal zero x(1) of the function F(x) = f(f(x)) - x, where f(x) = cos(x) - sin(x).

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A206291 Decimal expansion of the middle zero x(2) of the function F(x) = f(f(x)) - x, where f(x) = cos(x) - sin(x).

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A216863 The decimal expansion of the maximal zero x(3) of the function F(x) = f(f(x)) - x, where f(x) = cos(x) - sin(x).

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